fuzzy-anova

One-Way Analysis of Variance (ANOVA) for Triangular Fuzzy Data

A complete Python implementation of the fuzzy ANOVA framework based on:

Hsien-Chung Wu (2007), "Analysis of variance for fuzzy data", International Journal of Systems Science, 38(3), 235-246.

Developed by Harshit Trivedi and Dr. Masum Raj (Mathematics Professor).

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Overview

In traditional ANOVA, observations are crisp real numbers. In many real-world experiments—such as sensory evaluation, customer satisfaction surveys, and engineering tolerances—observations are inherently imprecise and naturally expressed as Triangular Fuzzy Numbers (TFN):

$$y_{ij} = (L_{ij}, \text{Core}{ij}, U{ij})$$

where:

• $L_{ij}$ is the lower bound of support (membership degree = 0)
• $\text{Core}_{ij}$ is the modal / core value (membership degree = 1)
• $U_{ij}$ is the upper bound of support (membership degree = 0)
• Condition: $L_{ij} \le \text{Core}{ij} \le U{ij}$

fuzzy-anova implements the complete Wu (2007) methodology, providing:

1. Alpha-Cut ($h$-level set) Method: Converts fuzzy numbers into real intervals at any membership degree $h \in [0, 1]$.
2. Lower & Upper Bound ANOVA Tables: Full ANOVA tables (SS, df, MS, F-statistic, P-value) calculated at any level-cut (Wu Table 2 & Table 3).
3. Quadratic Boundary Analysis: Computes exact quadratic coefficients $A \cdot h^2 + B \cdot h + C$ for both fuzzy boundaries.
4. Interval Sets $K_{H0}$ and $K_{H1}$: Determines the valid parameter ranges for accepting $H_0$ or $H_1$ via quadratic constraint optimization.
5. Degree of Belief Metrics:
   • Optimistic Degree $\lambda(K)$: Total interval measure / length of the supporting parameter set.
   • Pessimistic Degree $h_H$: Supremum of the opposing parameter set.
   • Confidence Degree: Overall balanced decision confidence $\min(\text{optimistic}, 1 - \text{pessimistic})$.
6. Zero Mandatory Heavy Dependencies: Fast pure-Python Snedecor's F-distribution fallback included, with automatic SciPy acceleration if SciPy is installed.

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Installation

pip install fuzzy-anova

Optional Dependencies

For faster F-distribution calculation using SciPy:

pip install "fuzzy-anova[scipy]"

For development and running tests:

pip install "fuzzy-anova[dev]"

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Quickstart

from fuzzy_anova import fuzzy_anova

# 1. Define groups with triangular fuzzy numbers (Lower, Core, Upper)
data = {
    "Design_1": [(9, 11, 13), (14, 16, 18)],
    "Design_2": [(11, 15, 19), (10, 15, 21), (11, 13, 15)],
    "Design_3": [(15, 18, 21), (14, 17, 20), (17, 20, 23)],
    "Design_4": [(15, 19, 23), (21, 24, 27)],
}

# 2. Run Fuzzy ANOVA at significance level alpha = 0.05
result = fuzzy_anova(data, alpha_sig=0.05)

# 3. Print the comprehensive formatted summary report
print(result.summary())

Accessing Result Quantities

# Decision and confidence metrics
print("Decision:           ", result.decision)            # 'H0' or 'H1'
print("Null Rejected:      ", result.is_null_rejected)     # True or False
print("Optimistic Degree:  ", result.optimistic_degree)   # e.g. 0.8808
print("Pessimistic Degree: ", result.pessimistic_degree)  # e.g. 0.0031
print("Confidence Degree:  ", result.confidence_degree)   # e.g. 0.8808

# Classical ANOVA core values (at h = 1)
print("SSTR:               ", result.sstr)
print("SSE:                ", result.sse)
print("F-Statistic:        ", result.f_stat)
print("Critical F-Value:   ", result.f_crit)
print("P-Value:            ", result.p_value)

# Boundary quadratic coefficients (A, B, C)
print("Lower Coeffs (A,B,C):", result.lower_coeffs.as_tuple())
print("Upper Coeffs (A,B,C):", result.upper_coeffs.as_tuple())

# Convert full results to a Python dictionary (ready for JSON serialization)
data_dict = result.to_dict()

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Alpha-Cut and Boundary Tables

You can evaluate explicit alpha-cuts or compute the lower/upper bound ANOVA tables at any specific cut level $h$:

from fuzzy_anova import get_alpha_cut, alpha_cut_anova

# 1. Get explicit interval endpoints [lower, upper] at h = 0.5
cuts = get_alpha_cut(data, h=0.5)
print(cuts["Design_1"])  # [(10.0, 12.0), (15.0, 17.0)]

# 2. Generate Table 2 (Lower Bound) & Table 3 (Upper Bound) at h = 0.0 (full support)
tables = alpha_cut_anova(data, h=0.0, alpha_sig=0.05)
print(tables.summary())

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Command-Line Interface (CLI)

fuzzy-anova provides a built-in CLI:

# Run the benchmark demonstration problem from Wu (2007)
fuzzy-anova --demo

# View version
fuzzy-anova --version

# Run ANOVA on your own dataset stored in a JSON file
fuzzy-anova data.json --alpha 0.05 --json

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API Reference

fuzzy_anova(groups, alpha_sig=0.05, alpha_cut_level=0.0, f_crit=None) -> FuzzyANOVAResult

Main function to perform One-Way Fuzzy ANOVA.

Parameter	Type	Default	Description
groups	Mapping or Sequence	required	Dictionary of {group_name: [(L, C, U), ...]} or list of lists [[(L, C, U), ...], ...].
alpha_sig	float	0.05	Statistical significance level $\alpha \in (0, 1)$.
alpha_cut_level	float	0.0	Cut level $h \in [0, 1]$ used for generating bound ANOVA tables.
f_crit	float or None	None	Optional critical value override (e.g. for matching textbook values).

get_alpha_cut(groups, h: float) -> Dict[str, List[Tuple[float, float]]]

Returns interval endpoints (lower_h, upper_h) for each observation at membership cut $h \in [0, 1]$.

alpha_cut_anova(groups, h=0.0, alpha_sig=0.05) -> AlphaCutANOVAResult

Computes the classical ANOVA table for both lower and upper bounds at level $h$.

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Verification & Testing

The package includes an extensive test suite verifying:

• Numerical accuracy against Wu (2007) published benchmarks to 4 decimal places
• Degenerate crisp numbers ($L = \text{Core} = U$) collapsing to classical ANOVA
• Identical groups accepting $H_0$ with confidence 1.0
• Widely separated groups rejecting $H_0$ with confidence 1.0
• Unequal sample sizes across treatments
• Asymmetric and overlapping fuzzy numbers
• Comprehensive input validation (types, shapes, NaN/Inf, ordering violations)

To run the automated tests:

python test_fuzzy_anova.py

or with pytest:

pytest

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Authors & Maintainers

• Harshit Trivedi — Software Architect & Core Implementation (GitHub)
• Dr. Masum Raj — Professor of Mathematics, Academic Mentorship & Theoretical Mathematical Verification

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Acknowledgements

Special gratitude to Dr. Masum Raj for mathematics mentorship, guiding the research on fuzzy hypothesis testing, and validating the numerical proofs and formulations from Wu (2007).

We also thank the peers and students who assisted with reviewing documentation, testing edge cases, and validating the outputs.

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References

1. Wu, Hsien-Chung (2007). "Analysis of variance for fuzzy data", International Journal of Systems Science, 38(3), pp. 235–246. DOI: 10.1080/00207720601157997.

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Citation

If you use fuzzy-anova in academic research, please cite both the foundational paper and this implementation:

@article{wu2007analysis,
  title   = {Analysis of variance for fuzzy data},
  author  = {Wu, Hsien-Chung},
  journal = {International Journal of Systems Science},
  volume  = {38},
  number  = {3},
  pages   = {235--246},
  year    = {2007},
  doi     = {10.1080/00207720601160822}
}

@software{fuzzy_anova_2026,
  author  = {Harshit Trivedi and Dr. Masum Raj},
  title   = {fuzzy-anova: One-Way Analysis of Variance (ANOVA) for Triangular Fuzzy Data},
  year    = {2026},
  url     = {https://github.com/AquillaRival/fuzzy-anova},
  version = {1.0.0}
}

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License

This project is licensed under the terms of the MIT License. Copyright (c) 2026 Harshit Trivedi, Dr. Masum Raj.